Estimating with uncertainty principle

[dsdsuster]

http://quantummechanics.ucsd.edu/ph130a/130_notes/node98.html

Can someone help me understand what's going on here?

He says "The idea is that the radius must be larger than the spread in position, and the momentum must be larger than the spread in momentum." which I suppose must be true in order to have a well-defined position and momentum? And then he uses the uncertainty of the momentum as a substitute for the actual momentum. This part in particular I don't really understand.

Thanks!

 

[RoyalCat]

I think it's because the average momentum is 0 (Closed orbit), so the time-average of the momentum squared (Which is what actually enters the equation), is exactly the variance in the expectation value of the momentum.